Average Error: 15.5 → 15.0
Time: 1.0m
Precision: 64
\[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{1}{\sqrt{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}} \cdot \left(\left(\frac{1}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}}} - \frac{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}}}\right) \cdot \frac{1}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}} \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}}}\right)\]
1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{1}{\sqrt{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}} \cdot \left(\left(\frac{1}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}}} - \frac{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}}}\right) \cdot \frac{1}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}} \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}}}\right)
double f(double x) {
        double r30033049 = 1.0;
        double r30033050 = 0.5;
        double r30033051 = x;
        double r30033052 = hypot(r30033049, r30033051);
        double r30033053 = r30033049 / r30033052;
        double r30033054 = r30033049 + r30033053;
        double r30033055 = r30033050 * r30033054;
        double r30033056 = sqrt(r30033055);
        double r30033057 = r30033049 - r30033056;
        return r30033057;
}
↓
double f(double x) {
        double r30033058 = 1.0;
        double r30033059 = 0.5;
        double r30033060 = x;
        double r30033061 = hypot(r30033058, r30033060);
        double r30033062 = r30033059 / r30033061;
        double r30033063 = r30033059 + r30033062;
        double r30033064 = sqrt(r30033063);
        double r30033065 = r30033058 + r30033064;
        double r30033066 = sqrt(r30033065);
        double r30033067 = r30033058 / r30033066;
        double r30033068 = cbrt(r30033066);
        double r30033069 = r30033058 / r30033068;
        double r30033070 = r30033063 / r30033068;
        double r30033071 = r30033069 - r30033070;
        double r30033072 = r30033068 * r30033068;
        double r30033073 = r30033058 / r30033072;
        double r30033074 = r30033071 * r30033073;
        double r30033075 = r30033067 * r30033074;
        return r30033075;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.5

    \[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]

  2. Simplified15.5

    \[\leadsto \color{blue}{1 - \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]
  3. Using strategy rm

  4. Applied flip--15.5

    \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}} \cdot \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt15.8

    \[\leadsto \frac{1 \cdot 1 - \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}} \cdot \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}{\color{blue}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}} \cdot \sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}}\]

  7. Applied *-un-lft-identity15.8

    \[\leadsto \frac{\color{blue}{1 \cdot \left(1 \cdot 1 - \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}} \cdot \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}\right)}}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}} \cdot \sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}\]

  8. Applied times-frac15.5

    \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \frac{1 \cdot 1 - \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}} \cdot \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}}\]

  9. Simplified15.0

    \[\leadsto \frac{1}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \color{blue}{\left(\frac{1}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} - \frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}\right)}\]
  10. Using strategy rm

  11. Applied add-cube-cbrt15.0

    \[\leadsto \frac{1}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \left(\frac{1}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} - \frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}{\color{blue}{\left(\sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}\right) \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}}}\right)\]

  12. Applied *-un-lft-identity15.0

    \[\leadsto \frac{1}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \left(\frac{1}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} - \frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \color{blue}{1 \cdot \frac{1}{2}}}{\left(\sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}\right) \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}}\right)\]

  13. Applied *-un-lft-identity15.0

    \[\leadsto \frac{1}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \left(\frac{1}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} - \frac{\color{blue}{1 \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} + 1 \cdot \frac{1}{2}}{\left(\sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}\right) \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}}\right)\]

  14. Applied distribute-lft-out15.0

    \[\leadsto \frac{1}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \left(\frac{1}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} - \frac{\color{blue}{1 \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right)}}{\left(\sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}\right) \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}}\right)\]

  15. Applied times-frac15.1

    \[\leadsto \frac{1}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \left(\frac{1}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} - \color{blue}{\frac{1}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}} \cdot \frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}}}\right)\]

  16. Applied add-cube-cbrt15.0

    \[\leadsto \frac{1}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \left(\frac{1}{\color{blue}{\left(\sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}\right) \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}}} - \frac{1}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}} \cdot \frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}}\right)\]

  17. Applied *-un-lft-identity15.0

    \[\leadsto \frac{1}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \left(\frac{\color{blue}{1 \cdot 1}}{\left(\sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}\right) \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}} - \frac{1}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}} \cdot \frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}}\right)\]

  18. Applied times-frac15.0

    \[\leadsto \frac{1}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \left(\color{blue}{\frac{1}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}} \cdot \frac{1}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}}} - \frac{1}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}} \cdot \frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}}\right)\]

  19. Applied distribute-lft-out--15.0

    \[\leadsto \frac{1}{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}} \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}} \cdot \left(\frac{1}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}} - \frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}}\right)\right)}\]

  20. Final simplification15.0

    \[\leadsto \frac{1}{\sqrt{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}} \cdot \left(\left(\frac{1}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}}} - \frac{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}}}\right) \cdot \frac{1}{\sqrt[3]{\sqrt{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}} \cdot \sqrt[3]{\sqrt{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}}}\right)\]

Reproduce

herbie shell --seed 2019120 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))